7.5. Calculate Yield Strength Programmatically#

# imports
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.stats import linregress
%matplotlib inline
%ls
ABS_HIPS_plot_with_error_bars.png
Al60601_raw_data.xls
Steel1018_raw_data.xls
bar_plot_with_error_bars.ipynb
calculate-yield-strength-programmatically.ipynb
images/
mechanical-properties-from-stress-strain-curves.ipynb
mechanical-properties-intro.md
plotting-stress-strain-curves.ipynb
stress-strain_curve.png
stress_strain_curve_with_inset.png
stress_strain_curve_with_inset_elastic_region.ipynb
df_al = pd.read_excel('Al60601_raw_data.xls')
df_al.head()
WARNING *** OLE2 inconsistency: SSCS size is 0 but SSAT size is non-zero
TESTNUM POINTNUM TIME POSIT FORCE EXT CH5 CH6 CH7 CH8
0 542 1 8.470 0.02256 201.146011 -0.001444 0.007552 NaN NaN NaN
1 542 2 8.632 0.02330 206.599442 0.000302 0.007552 NaN NaN NaN
2 542 3 10.027 0.02846 287.512573 0.003044 0.018898 NaN NaN NaN
3 542 4 11.031 0.03232 365.380981 0.009881 0.022061 NaN NaN NaN
4 542 5 11.928 0.03616 447.813965 0.014085 0.033652 NaN NaN NaN
df_steel = pd.read_excel('Steel1018_raw_data.xls')
df_steel.head()
WARNING *** OLE2 inconsistency: SSCS size is 0 but SSAT size is non-zero
TESTNUM POINTNUM TIME POSIT FORCE EXT CH5 CH6 CH7 CH8
0 523 1 6.189 0.07302 202.924728 0.000402 -0.028272 NaN NaN NaN
1 523 2 6.549 0.07396 205.714890 -0.000238 -0.034549 NaN NaN NaN
2 523 3 7.148 0.07624 217.763336 -0.000713 -0.030140 NaN NaN NaN
3 523 4 9.146 0.08438 316.306122 0.002377 -0.025968 NaN NaN NaN
4 523 5 10.041 0.08822 417.003357 0.003089 -0.024100 NaN NaN NaN
d = 0.506
r = d/2
A = np.pi*r**2

stress_al = (df_al['FORCE']/A)*0.001
strain_al = df_al['CH5']*0.01

stress_steel = (df_steel['FORCE']/A)*0.001
strain_steel = df_steel['CH5']*0.01

Plot the full stress strain curve#

fig,ax = plt.subplots()
ax.plot(strain_al, stress_al)
ax.plot(strain_steel, stress_steel)
ax.set_xlabel('Strain (in/in)')
ax.set_ylabel('Stress (ksi)')
ax.set_title('Stress-Strain Curve of Al6061 and Steel1018 in Tension')
ax.legend(['Al6061','Steel1018'])

plt.show()
../_images/6f9db4f5fd4b72caea3f208eb2154b21443d72528b24813c253bfa2f79198264.png

Caclulate elastic modulus#

fig,ax = plt.subplots()
ax.plot(strain_al, stress_al)
ax.plot(strain_steel, stress_steel)

ax.set_title('Inset of elastic region')
ax.set_xlabel('Strain (in/in)')
ax.set_ylabel('Stress (ksi)')
ax.legend(['Al6061','Steel1018'])

ax.set_xlim([0,0.01])
ax.set_ylim([0,70])

plt.show()
../_images/13ba3069c24fb057173ae21bb40b0291f3594f2af1504db4e335763850d339e2.png
# Find the elastic modulus of Al6061
# use stress and strain values from stress=0 to stress=35 ksi
linear_stress_al_mask = stress_al < 35
linear_stress_al = stress_al[linear_stress_al_mask]
linear_strain_al = strain_al[linear_stress_al_mask]

linear_regression_output = linregress(linear_strain_al, linear_stress_al)
E_al = linear_regression_output[0]

print(f'The elastic modulus of Al6061 is {round(E_al,1)} ksi')
The elastic modulus of Al6061 is 9909.4 ksi
# Find the elastic modulus of Steel1018
# use stress and strain values from stress=0 to stress=55 ksi
linear_stress_steel_mask = stress_steel < 55
linear_stress_steel = stress_steel[linear_stress_steel_mask]
linear_strain_steel = strain_steel[linear_stress_steel_mask]

linear_regression_output_steel = linregress(linear_strain_steel, linear_stress_steel)
E_steel = linear_regression_output_steel[0]

print(f'The elastic modulus of Steel1018 is {round(E_steel,1)} ksi')
The elastic modulus of Steel1018 is 40125.4 ksi

Cacluate Yield Strength#

The yield strength of a metal is calculated from a stress strain curve by drawing a line parallel to the elastic modulus that goes through the point strain=0.002. Where that line crosses the stress strain curve, that amount of stress is the yield strength

Point-Slope Formula for a line:

\[ y - y_1 = m(x-x_1) \]

Where \(m=\) slope in the linear elastic region (elastic modulus, E), \(x_1=0.002\), and \(y_1 = 0\)

Rewrite the equation, where \(y=stress_{offset}\) and the elastic modulus and value of \(x_1\) and \(y_1\) are included.

\[ stress_{offset} = E(strain-0.002) \]
# calculate the yield strength of Al6061
stress_offset_al = E_al*(strain_al-0.002)

fig, ax = plt.subplots()

ax.plot(strain_al,stress_al,strain_al,stress_offset_al)
ax.set_ylim([0,70])
ax.set_xlim([0,0.01])

plt.show()
../_images/9a444552cd798adc82c869eec8babb794fad688e51a4dd0fd6d2d41d119205c8.png
YS_al = 41.3653 # by zooming into the plot that contains the stress strain curve of Al6061 and the offset stress
print(f'The yield strength of Al6061 is {round(YS_al,1)} ksi')
The yield strength of Al6061 is 41.4 ksi
# Calculate the yield strength of Steel1018
stress_offset_steel = E_steel*(strain_steel-0.002)

fig, ax = plt.subplots()

ax.plot(strain_steel,stress_steel,strain_steel,stress_offset_steel)
ax.set_ylim([0,70])
ax.set_xlim([0,0.01])

plt.show()
../_images/0f4148b826d1e7d3c7c68a7573073c9e4fdad0a381cced6adac138ce1fde087f.png
YS_steel = 57.4495 # by zooming into a plot showing the stress strain curve of steel1018 and the offset stress
print(f'the yield strength of steel1018 is {round(YS_steel,1)} ksi')
the yield strength of steel1018 is 57.4 ksi

Calculate yield strength programatically#

f = np.array(stress_al)
g = np.array(stress_offset_al)
yield_stress_index = np.argwhere(np.diff(np.sign(f - g)))[0][0]
yield_strength_al = stress_al[yield_stress_index]

fig, ax = plt.subplots()

ax.plot(strain_al,stress_al, "*")
ax.plot(strain_al, stress_offset_al, "." )
ax.plot(strain_al[yield_stress_index],yield_strength_al,'go')

ax.set_ylim([0,50])
ax.set_xlim([0,0.01])

plt.show()
../_images/cfcefe67a11250c973fa3986867d85cd5f3ebd8938d2091553b5f67e3399f0bb.png
print(f'The yield strength of Al6061 found programatically is {round(yield_strength_al,1)} ksi')
The yield strength of Al6061 found programatically is 41.3 ksi
# Find the yield strength of steel programatically

f = np.array(stress_steel)
g = np.array(stress_offset_steel)

yield_strength_steel_index = np.argwhere(np.diff(np.sign(f - g)))[0][0]
yield_strength_steel = stress_steel[yield_strength_steel_index]

fig, ax = plt.subplots()

ax.plot(strain_steel,stress_steel)
ax.plot(strain_steel,stress_offset_steel)
ax.plot(strain_steel[yield_strength_steel_index],yield_strength_steel,'go')

ax.set_ylim([0,70])
ax.set_xlim([0,0.01])


plt.show()
../_images/1af1c4478109f09c31dfc0382e5353599da9663814fc5d5cdd357b920cef5e3d.png
print(f'The yield strength of Steel1018 found programatically is {round(yield_strength_steel,1)} ksi')
The yield strength of Steel1018 found programatically is 57.6 ksi

Write a function to calculate the intersection of two lines#

# write a function to calculate the intersection of two lines defined by 4 points

def intersection_point(Ax1, Ay1, Ax2, Ay2, Bx1, By1, Bx2, By2):
    d = (By2-By1)*(Ax2-Ax1)-(Bx2-Bx1)*(Ay2-Ay1)
    if d:
        uA = ((Bx2-Bx1)*(Ay1-By1)-(By2-By1)*(Ax1-Bx1))/d
        uB = ((Ax2-Ax1)*(Ay1-By1)-(Ay2-Ay1)*(Ax1-Bx1))/d
    else:
        return None
    if not(0 <= uA <= 1 and 0 <= uB <= 1):
        return None
    x = Ax1 + uA * (Ax2 - Ax1)
    y = Ay1 + uA * (Ay2 - Ay1)
    
    return x, y
strain_steel_array = np.array(strain_steel)
stress_steel_array = np.array(stress_steel)
strain_al_array = np.array(strain_al)
stress_al_array = np.array(stress_al)

Caclulate yield strength using the intersection function#

first = yield_strength_steel_index
second = first + 1
# A points from the stress strain curve
Ax1 = strain_steel_array[first]
Ay1 = stress_steel_array[first]
Ax2 = strain_steel_array[second]
Ay2 = stress_steel_array[second]
# B points from the offset line
Bx1 = strain_steel_array[first]
By1 = stress_offset_steel[first]
Bx2 = strain_steel_array[second]
By2 = stress_offset_steel[second]

# run our function that finds the intersection point
x, y = intersection_point(Ax1,Ay1,Ax2,Ay2,Bx1,By1,Bx2,By2)

print(x,y)
0.0034317318641272137 57.448780022628426
fig,ax = plt.subplots()


ax.plot(strain_steel_array,stress_steel_array)
ax.plot(strain_steel_array,stress_offset_steel)
ax.plot(x,y,'go')

ax.set_ylim([0,70])
ax.set_xlim([0,0.01])

plt.show()
../_images/287950225ffc5d56239863193e21f8bec247a79baa46e6d1cf6cf596381f6f24.png
print(f'The yield strength of Steel1018 calculated programmatically is {round(y,1)} ksi')
The yield strength of Steel1018 calculated programmatically is 57.4 ksi